Question #20: What does the cable equation mean for neurons?

A simplified explanation of capacitance in neuronal membranes is that higher capacitance will tend to will cause a flow of ions towards the membrane on the cytoplasmic side due to the difference in charge across the membrane, called a displacement current; https://en.wikipedia.org/wiki/Cable_theory#/media/File:NeuronCapacitanceRev.jpg

– Cable theory can be derived in part from Ohm’s law, the fundamental theory of electricity that models the current flowing between two points as equal to the voltage distance between the two points divided by the material’s resistance, or in other words, the classic equation V = IR.

– The greater the cross-sectional area of the neurite’s cytosol (the interior part of it, containing biomolecules, electrolytes, and other ions), the easier an ion can flow through it, so the neurite’s longitudinal resistance, r_l, will be lower.

– If the cell membrane is more resistant to ion flow into or out of the cell (due to high membrane resistance, r_m), then charge will tend to accumulate inside the cell, and it will have a higher ionic current flowing down longer distances in the neurite. This is often represented by a paremeter called the length current, λ, equal to the square root of r_m divided by r_l.

– If a cell membrane has a lower membrane capacitance (c_m, which is usually a fairly constant value), then the relative ion flow down the neurite will be greater, due to a lower displacement current. How quickly the membrane voltage changes in response to a current injected at at given point can be predicted by the time constant, τ, equal to the product of c_m times r_m.

– An electrotonic potential results from a local change in ion conductance, e.g. after a synaptic event, that does not propagate. It becomes exponentially smaller as it spreads. This is opposed to an action potential, which reaches the voltage threshold by which it does propagate down the neurite (due to the opening of voltage-gated ion channels), and then spreads like a wave.

– Dendritic trees can perform non-linear integration of signals that can be predicted on the basis of cable theory. The existence of subthreshold membrane potential fluctuations in dendrites, which based on my understanding should dominate neuronal signaling, can allow variations in synaptic weight distributions and input timing to encode a substantial amount of information within a single neuron.

Inspired by CalTech’s Question #20 for cognitive scientists: “Derive the cable equation (for a uniform cylinder, with optimal boundary conditions). What does it mean for neurons?”

Question #19: Ion channel biophysics

What are the biophysics of voltage-gated sodium channels? 

Sodium channels are a major component of excitable membranes. They are an intrinsic component of excitable tissue that allows them to generate and propagate action potentials. These electrical signals are essential for proper neuronal communication.

The channel looks like a barrel, with 4-fold symmetry, and a diameter of about 10 nm. The channel has an activation gate, through which sodium ions can flow through. If the activation gate is closed, no ions can pass through, but if it is open, ions can pass through the pore. The channel is closed at rest, wherein the membrane voltage potential is polarized. When a sufficient voltage depolarization across the membrane occurs, the membrane will draw the gates open, allowing sodium ions to flow through and leading to further depolarization. When enough sodium has passed, the further voltage change causes the inactivation gate to close, thus stopping the flow of sodium ions and leading to repolarization.

Sodium channels are selective for sodium ions because the inner filter of the pore is highly negatively charged; the Na+ ion has a positive charge and will bind well to the inner filter. K+ ions, while also positively charged, cannot pass through because of a size restriction. The gate is not large enough for them to fit through. For ions to pass, they need to be smaller than the diameter of the filter; for ions with a larger diameter to pass, the filter would need to enlarge; however, the size of the filter cannot increase because the pore has a fixed size. These are the unique properties of the sodium channel that allow it to selectively conduct sodium.

Sodium channels are good targets for many drugs and toxins. For example, tetrodotoxin specifically binds to voltage-gated sodium channels and can stop sodium channels from opening, thereby blocking all neural signaling. 

What are the biophysics of transmitter-gated channels? 

Transmitter-gated channels are opened by transmitters. They are then generally ion-selective. To open the channel, the transmitter needs to bind to the receptor. The transmitter binding causes an allosteric change that allows another part of the channel to open, known as the ion channel gate. When open, the ion channel gate allows specific ions to pass through.

A special example is the NMDA receptor. Under normal circumstances, the NMDA receptor is blocked by Mg2+ and Zn2+ ions. When the post-synaptic neuron is depolarized, however, Mg2+ and Zn2+ ions are repelled. In this case, the receptor can be activated by glutamate. When activated, the NMDA receptor allows positive ions to pass through (K+, Na+, and Ca2+ ions), which can help sustain depolarization and lead to intracellular signaling events such as long-term potentiation.

NMDA receptors are often called “coincidence detectors” because these two events must occur together for the channel to open. First, the NMDA receptor must be activated by the post-synaptic being depolarized, and second, glutamate must be released.

Another example is the nicotinic acetylcholine receptor. When acetylcholine binds to the receptor, the channel opens. This allows sodium and potassium ions to pass through, which leads to depolarization and therefore a neural signal.

Most types of ion channel activity in the brain need regulation. Regulation can occur post-translationally through the addition of a phosphate group to one or more amino acids. The addition of a phosphate group to a particular location of the AMPA receptor, for example, has been shown to increase the probability of AMPA channel opening. The Ca2+/calmodulin kinase II pathway is able to phosphorylate the GluA1 AMPA receptor subunit at Ser831, causing an increase in AMPA channel conductance.

In addition to the post-translational regulation of channel activity, many channels are regulated by endogenous compounds in the brain. Serotonin is a monoamine neurotransmitter that regulates various types of sodium channels and potassium channels. Dopamine is also a monoamine neurotransmitter, and it can be found in extrasynaptic regions. Dopamine has been shown to increase potassium channel activity by activating dopamine D1 receptors in axons.  

Together, the biophysics of ion channels allow for neural signaling by allowing for the passage of ions into and out of the cell. This allows for changes in membrane potential and intracellular signaling. 

Inspired by CalTech’s Question #19 for cognitive scientists: “Describe the main biophysical characteristics of ionic channels. How does its biophysical properties contribute to its physiological function? What is thought to be the basis for the channels ion selectivity?”

Microwave irradiation as a brain banking tool

I love random reading old papers. They’re like treasure chests into secret knowledge that others might overlook. Here’s one: Guidotti et al 1974, “Focussed microwave radiation: a technique to minimize post mortem changes of cyclic nucleotides, dopa and choline and to preserve brain morphology”.

As a summary, they found that microwave irradiation in rat and mouse brains for 2 seconds led to the total inactivation of the enzymes involved in the regulation of numerous brain metabolites, such as cyclic AMP, cyclic GMP, choline acetyltransferase, and phosphodiesterase. As a result, it allowed for the accurate dissection of different brain nuclei for measurement of the concentrations of a variety of different metabolites in a variety of brain areas.

Rapid inactivation of enzyme activity; https://doi.org/10.1016/0028-3908(74)90061-6

The microwave irradiation method described by Guidotti et al. 1974 was relatively crude, as the authors only described microdissection of brain nuclei. It is unclear whether delicate cellular features such as synapses might also be preserved.

Interestingly, microwave irradiation produced extensive denaturation of proteins throughout the brain. Because since enzymes control metabolism, the inactivation of these enzymes can minimize changes in metabolism.

Microwave radiation could theoretically be combined with other brain preservation methods, such as fixation or cryopreservation, to minimize enzyme-driven autolysis during the procedure. A potential limitation of this method is that microwave heating is limited by the thermal conductivity of the tissue.

What can super-resolution light microscopy tell us about biomolecular brain preservation?

A lot. A nice example of the power of aldehyde fixatives to preserve fine molecular detail is Helm et al 2021.

Dendritic spines, which are often considered the functional units of neuronal circuits, strongly vary in size and shape. This study used electron microscopy, super-resolution microscopy, and quantitative proteomics to characterize > 47,000 spines at > 100 synaptic targets, helping to quantify variation in biomolecular composition across spines. Their study is amazing in part because of their technical advances, which allow for the beautiful visualization of biomolecules across neuronal membranes.

People often say that connectomics is not enough for brain information preservation because each dendrite has its own spread of ion channels. This distribution of ion channels will tell you whether a dendritic spike will occur, which is incredibly important to figure out synapse function.

If the local dendritic tree goes over a certain threshold of depolarization, then the local ion channels will open up and amplify what would have come in with the synapses alone. This also could synergize with clusters of synapses.

Theoretically, each neuron could have a unique spread of ion channels along dendrites, which could potentially make synaptic connectivity data alone insufficient, even if you have or can accurately infer synapse molecular information.

It’s hard to find examples of real evidence in the literature for or against how important these effects are. But the nonlinear effect of clusters of synapses is something that we potentially can’t account for with electron microscopy data alone.  

This is a reasonable/principled objection to the idea of brain information preservation via connectivity. Personally, I find it quite plausible. A way to address this objection is to say that super-resolution microscopy techniques like those used by Helm et al 2021 could be applied to decoding memories from fixed brain tissue via measuring biomolecules, without necessarily assuming that synapses alone will be sufficient.

Biophysically realistic neuron modeling in layer four of visual cortex

I recently checked out the interesting article by Arkhipov et al 2018 and wanted to discuss it here. They use a well-grounded computational model of L4 (which is the input layer) of mouse visual cortex that is capable of replicating a number of experimental observations. Their model combines biophysically detailed neuron models, synaptic dynamics, and experimentally constrained connectivity. Here is their summary figure describing the biophysical model as well as the leaky integrate and fire (LIF) portion:

https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1006535

This model is effectively supposed to summarize much of what is known in the entire field. It is in line with the Markram approach to brain modeling: we know that from electrophysiology data how these subcircuits of V1 L4 cells work and now let’s use this prior to knowledge to build a Hodgkin-Huxley based model of it. They assessed their model performance by reproducing a number of experimental findings, such as:

1. They reproduced the statistical features of V1 neuronal responses, such as their log-normal distribution.

2. They systematically investigated how neurons in the model respond to a variety of visual stimuli, both the type of stimuli mice might see in the real world (movies) and ones they would not (gratings).

3. As expected from previous literature, they showed that connectivity rules strongly impact neuronal responses. For example, adding recurrency to the network not only amplifies and synchronizes firing rates, but also biases the neuronal tuning properties.

While not directly studying memory storage, the Arkhipov et al study is relevant to the brain preservation problem in a number of ways.

Primarily, it shows that compartmentalized circuits, such as L4 of V1 can be simulated in accurate ways using contemporary biophysical models.

It also highlights the likely importance of connectivity rules in memory storage. Connectivity patterns form the backbone of any computational model, and Arkhipov et al demonstrated how connectivity rules can shape (and constrain) network activity. Therefore, this is another data point that we must consider connectivity patterns to be crucial factors in memory storage.

Their study used compartment models (“compartmental representation of somato-dendritic morphologies (~100–200 compartments per cell) and 10 active conductances at the soma that enabled spiking and spike adaptation”). Their results are a data point that at the single-neuron level, this amount of information may be sufficient for simulations able to reproduce in vivo functional properties, therefore suggesting a potentially reduced need for fine-scale detail preservation, although this is of course still subject to considerable uncertainty.

They also compared the performance of their model to a much simplified version, where biophysical neuron models were replaced by point-neuron models with either instantaneous or time-dependent synaptic kinetics. They found that the biophysically realistic model had quantitatively better accuracy compared to the experimental model, although even with extreme simplification, the model still performed fairly well. This suggests that a level of model detail even above the compartment models may be sufficient.

While they got their connectivity patterns in a random fashion, one might imagine instead getting connectivity data from an electron microscopy-based connectomics data set. It would be interesting to see if, in a realistic biophysical model and a realistic connectomics data set, they could still reproduce a similar set of functional observations.

Theoretically, using these kinds of models, if one were to go beyond just L4 of V1 to a whole brain, it is interesting to think what kind of functional properties might emerge. However, I personally think we should be judicious about building such models.

(Thanks to Ken Hayworth for a discussion about this paper.)

Towards building an accurate brain molecular concentration database

An interesting study by Shichkova et al 2021, who perform proteomic/metabolomic profiling studies in different brain areas and cell types, integrate and normalize the data, and generate a Brain Molecular Atlas database. They then use this database to create more accurate representations of biomolecular systems that are simulation-ready.

An accurate molecular concentration database is a prerequisite for creating data-driven computational models of biochemical networks. The Brain Molecular Atlas that they present overcomes the obstacles of missing or inconsistent data to support systems biology research as a resource for biomolecular modeling.

Highly expressed protein networks in different cell types; https://www.frontiersin.org/articles/10.3389/fnmol.2021.604559/full

One way this is relevant to brain preservation is that we will need accurate molecular concentrations to build realistic simulations of brain networks and map engrams. This is because engrams are likely composed of many molecular species and pathways that need to be accurately modeled in their concentrations in order to create an accurate representation of the engram.

Engrams could be distributed across multiple brain regions and cell types, and likely have a large number of pathways involved. Accurate molecular concentrations in these different contexts would be essential to be able to map engrams without potential gaps or inaccuracies.

Human brain tissue can be effectively analyzed via electron microscopy at postmortem intervals up to 100 hours if the body is stored at cold temperature after death

I recently saw this interesting quote from Kay et al 2013 in their Nature Protocols article:

For tissue preparation, we have incorporated array tomography and EM preparations into routine brain bank collection. We have managed to conduct very effective EM studies on tissues retrieved from donors with long post mortem intervals, up to 100 hours. In our experience a key element in tissue preservation for ultrastructure analysis is post mortem cold storage of the cadaver, with cold storage in a mortuary of around 4–6°C significantly reducing structural degradation.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3712649/

A neural pacemaker of aging?

Here is an interesting grant from Karl Deisseroth and Anne Brunet that I saw on NIH Reporter. It will be very interesting to see the results from these experiments in a few years.

Aging is a gradual process that results in the loss of cellular function across the body, leading to numerous chronic diseases that promote mortality. Elucidating the precise mechanisms of aging is critical for reducing illness and extending healthy lifespan. However, almost every tissue in the body is modified by aging, making it difficult to pinpoint the principal controller of aging. The goal of this proposal is to determine whether the brain modulates aging through coordinated activity patterns within discrete neuronal networks. We will use one of the shortest-living vertebrates, the African turquoise killifish, as a rapid, high-throughput model of aging to uncover genetically- defined neurons that regulate cellular metabolism and lifespan. Employing large-scale light-sheet imaging in killifish, we will visualize brain-wide calcium activity dynamics to unbiasedly identify neurons that respond to longevity interventions. We will characterize the genetic profiles of the identified neurons via a combination of immunohistochemical, single cell, and phosphorylated ribosome capture approaches. To examine whether these neurons play a causal role to control overall cellular function in the brain and other tissues, we will optogenetically activate these neurons and measure molecular signatures of youth and in vivo metabolic activity in the brain and peripheral tissues. We will monitor and manipulate neural activity throughout the short lifespan of killifish using fiber photometry to determine if this ‘neural pacemaker’ dictates the tempo of aging and youthful behavior. These approaches will then be extended to longer-lived species – zebrafish and mice. Knowledge resulting from these studies should be transformative to understand the fundamental mechanisms that regulate and synchronize aging and longevity. As age is the prime risk factor for many diseases, including neurodegenerative diseases, this proposal should provide new, circuit-based approaches to treat these diseases.

https://reporter.nih.gov/search/hKEP095-sESXSOTnzu5o7Q/project-details/10207466

What would “memory decoding” in the MICrONS data set imply?

Attention conservation notice: Not an area of expertise for me. Posted as in the spirit of Cunningham’s Law.

The recently posted MICrONS data set has functional imaging on 75,000 pyramidal neurons and EM-level anatomic data on 120,000 neurons.

Layer 2/3 cells from the MICrONS data set; screenshot from https://ngl.microns-explorer.org/

As the preprint describes: 

“The volume was imaged in vivo by two-photon microscopy from postnatal days P75 to P81 in a male mouse expressing a genetically encoded calcium indicator in excitatory cells, while the mouse viewed natural movies and parametric stimuli. At P87 the same volume was imaged ex vivo by serial section EM. Because the light and electron microscopic images can be registered to each other, these primary data products in principle contain combined physiological and anatomical information for individual cells in the volume, with a coverage that is unprecedented in its completeness.”

https://www.biorxiv.org/content/10.1101/2021.07.28.454025v2

As far as I can tell, it is an open question the degree to which the serial section EM can be used to predict the functional imaging data in this data set. But one might imagine that the EM data or wiring diagram could be eventually used to identify ensembles of neurons that respond to specific visual stimuli. If this could be shown, would it count as memory decoding? 

One of my neuroscience professors in grad school, Matthew Shapiro, spoke of “memory in the everyday sense of the word.” If we were to go up to a layperson and tell them that we had identified functional neuronal ensembles based on anatomic EM data to a sufficient degree of accuracy, they would probably not think that this meant that we had decoded a memory.

However, maybe they would be not be appreciating the eventual implications. Perhaps this ultimately is the core of what will eventually be required for memory decoding in the everyday sense. To me, this seems like somewhat of an open theoretical question.

Synapses can be seen and analyzed via electron microscopy at 2-4 days postmortem in at least some cases

Many people will say that the brain decomposes completely within minutes after death. However, they usually don’t offer data when they make such claims.

My impression from reading the literature is that actual postmortem decomposition is slower than many people think.

Here’s an example from a recent article I read, Henstridge and colleagues 2015. They studied a brain tissue that they banked in part via immersion fixation in 4 % paraformaldehyde and 2.5 % glutaraldehyde in 0.1 M PB for 48 hours.

From Table 1, here is the pathoclinical information, including the postmortem interval. As you can see, some of the postmortem intervals prior to preservation are >3 days. It’s unclear to me if the body/brain was refrigerated during this postmortem interval, but it’s a likely possibility.

Table 1; doi: 10.1186/s40478-015-0232-0

As a side note, this is a fascinating data set that includes intelligence test scores at age 11. This allows the researchers to adjust for premorbid cognitive functioning in a robust way as they investigate the causes of age-related cognitive decline. 

After preparing the tissue for electron microscopy, they found that they were able to study synapses. Only a small percentage of the identified presynaptic and postsynaptic terminals were found to have degenerating profiles. This seems to have been attributed to antemortem Alzheimer’s disease, rather than postmortem decomposition.

Figure 13; 10.1186/s40478-015-0232-0

There are all sorts of confounds here. The study doesn’t seem to have been focused on identifying the limits of the postmortem interval in which this sort of study is still informative. They may have adjusted for the postmortem interval and presence of decomposed synapses in some sort of way. Et cetera.

But generally speaking, this data tells us that it’s reasonable to think that synapses might still be largely structurally intact even at 2-4 days postmortem interval. At least in some cases depending on the cause and circumstances of death. This is actual data, rather than pure speculation.